Question

what is the nearest point of X^4+y^4=1 to x+8y=24

Answer 1

is this for calculus?

Answer 2

you have to find some slope that are perpendicular and measure the distance between them im assuming

Answer 3

lol ... well, the line only has 1 slope for every point :)

Answer 4

use distance formula and minimize it by takin derivative..

Answer 5

slope of the line = -1/8

determine when the detrivative of the other = 1/8

Answer 6

...dropped me - sign :)

Answer 7

Dx{x^4+y^4=1}

4x^3 + 4y^3 y' = 0

y' = (-4x^3)/(4y^3)
= -x^3/y^3

Answer 8

there was some sort of bacterial infestation in the ham last night and when we woke up this morning this is what we found lol

Answer 9

-1 -x^3
--- = ------ ; got to test all the possibles
8 y^3

-x^3 = 1
x^3 = -1
x = -1

but then again, the negative can go south to the 8 sooo

x^3 = 1 when x=1 ... what am i missing in this if anything?

Answer 10

...at any rate I see combos of x=1 and y=2 ... just gotta sort out the signs :)

Answer 11

we want to match these points to a line that is perp to our slope to get the shortest distances ... so the line of y = 8x -8(Px) +Py

Answer 12

Maybe x is 1/2....

Answer 13

\[-\left(\frac{x}{y}\right)^3=-\frac{1}{8}\]

(-1,-2) and (1,2) is what I see ....

Answer 14

but if those are points on the graph where the sloep of the tangent line = -1/8, i have to mention that those points aren't *on* the graph of x^4 + y^4 = 1... =/

Answer 15

y1 = 8x -8(-1) -2
= 8x +6

y2 = 8x -8(1) +2
= 8x -6

perhaps?

Answer 16

hmmm ... that could be an issue; but never sweat the small stuff :)

Answer 17

lol

Answer 18

1/ 1/2 is 1/8

Answer 19

Nah, can't be right..

Answer 20

http://www.wolframalpha.com/input/?i=x^4%2By^4%3D1+and+x%2B8y%3D24

the graphs look like this

Answer 21

and its the top of the skinny eclipse thats gonna be our breadwinner

Answer 22

can we zoom in on it more?

Answer 23

http://www.wolframalpha.com/input/?i=x^4%2By^4%3D1+and+x%2B8y%3D24%2C+x+%3D+-5+to+5%2C+y+%3D+0+to+4

Answer 24

nah; but its centered at the origin

Answer 25

lol ... matbe we can :)

Answer 26

http://www.wolframalpha.com/input/?i=x^4%2By^4%3D1+and+x%2B8y%3D8.35

getting closer ;)

Answer 27

ok...let's step back for a second...

-x^3/y^3=-1/8
x^3/y^3=1/8
x/y=1/2
y=2x

Answer 28

this means that the slope of the tangent line will be -1/8 when y is 2x.

plug this back into the original...

Answer 29

you get x= as the 4th root of 1/17

Answer 30

y would then be 2 times the fourth root of 17.

Answer 31

and these values work for the original equation.

Answer 32

That's lookin good..

Answer 33

very ugly, but i bet if you throw a line through this point with slope fo 8, it'll be a nice perpendcular to the line

Answer 34

lol...i'd use point slope form on the graphing calculator =)

Answer 35

http://www.wolframalpha.com/input/?i=x^4%2By^4%3D1+and+x%2B8y%3D8.35

i got an accuracy of a few decies :)

Answer 36

http://www.wolframalpha.com/input/?i=x%5E4%2By%5E4%3D1+and+x%2B8y%3D24+and+y-2%281%2F17%29%5E.25%3D8%28x-%281%2F17%29%5E.25%29%2C+x+%3D+0+to+4%2C+y+%3D+0+to+4

Answer 37

so now i guess we'd need to find the intersection of\[x+8y=24\]\[y-2(1/17)^{1/4}=8[x-(1/17)^{1/4}]\]

Answer 38

then calculate the distance between the points...

who came up with this problem?!?

Answer 39

dante im guessing

Answer 40

well, it doesn't seem to get any better, because the x-coordinate alone is\[x=(24-48\sqrt[4]{17})/65\]
I'm not sure my brain can take anymore...

Answer 41

my mistake...24+48...

Answer 42

and inside the radical is 1/17..

Answer 43

My 3d soft has to make this sort of calculation all of the time (collision avoidance).

Have to see if I can find how they do it....